The 32-year-old Croatian has been without a club since leaving the Cottagers in the summer after his one-year contract expired.
West Ham United striker Andy Carroll has re-injured his right foot after returning to training, the Premier League club said on Monday.
Downing made 91 appearances for Liverpool in his two years at the club, where he won the 2012 League Cup.
Boulahrouz, who moved to Lisbon last season could cut short his stay due to the Portugal side's financial woes.
After completing the deal for Andy Carroll, the Hammers management are anxiously waiting for Mourinho's 25-member Chelsea squad.
The towering England striker has now completed a permanent switch to the east London club on a six-year deal, with West Ham having the option of a further two years.
Manchester City responded to losing their Premier League title with a comfortable 2-1 victory over West Ham United.
Manchester United were forced to twice come from behind to draw 2-2 at West Ham delaying their charge to the Premier League title.
Also in the mid-week fixtures, Man City host relegation-threatened Wigan and Chelsea play at neighbours Fulham.
The draw left Liverpool seventh, a place below what is set to be the final Europa League qualification place.
The stadium will have a 54,000 capacity once it is converted for use by the Premier League club, who have been granted a 99-year lease.
Andy Carroll scored just his second goal for West Ham to secure a 1-0 win over Swansea.
Allardyce criticised referee Phil Dowd after West Ham were denied what he considered a penalty while United were awarded a spot kick.
Arsenal ran riot with four goals in 12 mad minutes after the break having fallen behind in the first half to Jack Collison's goal.
Arsenal striker Theo Walcott has described his side's home game against West Ham United as "massive".
Manchester United and Arsenal won 1-0 in replays to reach the fourth round of the FA Cup.
Chelsea have recalled Israeli midfielder Yossi Benayoun from his loan spell at West Ham United.
James Morrison was twice denied by the crossbar as West Bromwich Albion were forced to settle for a 0-0 draw.
Andy Carroll will be out of action for up to eight weeks with a knee injury.